$A_n^{(1)}$-Geometric Crystal corresponding to Dynkin index $i=2$ and its ultra-discretization

Abstract: Let $g$ be an affine Lie algebra with index set $I = {0, 1, 2,..., n}$ and $g^L$ be its Langlands dual. It is conjectured that for each $i \in I \setminus {0}$ the affine Lie algebra $g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for $g^L$. We prove this conjecture for $i=2$ and $g = A_n^{(1)}$.